First, \(\pi _\lambda \)  is a surjective submersion; the Implicit Function Theorem provides local eigenvalues, and so local sections, through any point of \(\mathrm {Spec}(V)\) . By dimension count, \(\pi _\lambda \)  is a local diffeomorphism. As \(\pi _\lambda \)  is also proper, it is a covering map [1]}. Then, as each fiber has exactly \(n\)  elements, \(\pi _\lambda \)  is an \(I_n\) -bundle. Indeed, a local trivialization is a map of the form
\( \begin{split}\phi \colon U \times I_n &\rightarrow \mathrm {Spec}(V)|_U\\(A,i) &\mapsto (A,\lambda _i(A))\end{split}\) 
